Moments and Fourier transforms of B-splines

The problem of determining the moments and the Fourier transforms of B-splines with arbitrary knots is considered. There exists a simple connection between the moments of such splines and the so-called extended Stirling numbers of the second kind which are defined in section 2. Some recurrence relations for the moments of B-splines with arbitrary knots are given in section 3. In the case of equidistant knots we have also further recurrences. For the forward, central and perfect B-splines the explicit formulas for the moments are given in section 3. The Fourier transforms of B-splines is treated in section 4. The final section is devoted to so-called Stieltjes series connected with the nonnegative weight function w(x) and such that $\text{∫}\text{a}\text{b}\text{w(x) dx}\text{> 0}$ in some closed interval a, b]. It is proved that such series for the particular values of the independent variable may be expressed by the finite sums which contain the nodes and coefficients of the optimal (in the Davies sense) quadrature formulas.